Scilab Function
Last update : 24/3/2004

lsq_splin - weighted least squares cubic spline fitting

### Calling Sequence

  `[y, d] = lsq_splin(xd, yd [, wd], x)`

### Parameters

- `xd, yd`: vectors of the same size, datas to be fitted by a cubic spline
- `wd`: (optionnal) a vector of same format than `xd` and `yd`, weights of the least square fit.
- `x`: a strictly increasing (row or column) vector, breakpoints of the cubic spline
- `y, d`: vectors of same format than `x`, the triplet (x,y,d) defines the approximated cubic spline.

### Description

This function computes an approximated cubic spline s for the datas `xd, yd, wd` (in the following m is supposed to be the length of these vectors) and from a choice of the spline breakpoints `x` (for instance if you want n breakpoints uniformly choosen you may use `x=linspace(min(xd),max(xd),n))`). If S is the space of all cubic splines functions with breakpoints x1 < x2 < ... < xn then the resulting spline s is such that:

         _m_                               _m_
         \                           2     \                           2
         /   wd(k) (s(xd(k)) - yd(k))  <=  /   wd(k) (f(xd(k)) - yd(k))
         ---                               ---
         k=1                               k=1

for all f in S, i.e. realizes the minimum of the sum of the squared errors over all functions of S.

The spline s is completly defined by the triplet `(x,y,d)` (`y` and `d` are the vectors of the spline ordinates and first derivatives at the xi 's : yi=s(xi) and di=s'(xi)) and its evaluation at some points must be done by the `interp` function.

### Remarks

When `wd` is not given, all the points have the same weight 1.

A point (xd(k),yd(k)) is considered in the fit if xd(k) in [x1,xn] and wd(k) > 0. In particular you can put a null (or even negative) weight to all data points you want to ignore in the fitting. When the total number of points taken into account in the fit procedure is (strictly) less than 4 an error is issued.

The vector `xd` do not need to be in increasing order.

Depending on the number and on the positions of the xd(k) 's and on the choice of the x(i) 's there may be several solutions but only one is selected. When this occurs a warning message is displayed in the Scilab command window. This function is intended to be used when m is much larger than n and in this case no such problem may occured.

### Examples

    // this is an artifical example where the datas xd and yd
    // are build from a perturbed sin function
    a = 0; b = 2*%pi;
    sigma = 0.1;  // standard deviation of the gaussian noise
    m = 200;       // number of experimental points
    xd = linspace(a,b,m)';
    yd = sin(xd) + grand(xd,"nor",0,sigma);

    n = 6; // number of breakpoints
    x = linspace(a,b,n)';

    // compute the spline
    [y, d] = lsq_splin(xd, yd, x);  // use equal weights

    // plotting
    ye = sin(xd);
    ys = interp(xd, x, y, d);
    xbasc()
    plot2d(xd,[ye yd ys],style=[2 -2 3], ...
           leg="exact function@experimental measures (gaussian perturbation)@fitted spline")
    xtitle("a least square spline")
    xselect()

### See Also

`interp`, `splin`,

### Authors

  C. De Boor, A.H. Morris (code from the NSWC fortran lib)
  B. Pincon (scilab interface and slight modifications)